Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 42 / 1

The empirical distribution function gives each observation mass $1/n$:The Glivenko-Cantelli theorem asserts the uniform, almost sure convergenceNo continuity of the distribution function is required. Here is a proof that also handles its jumps. Let $U_i$ be independent uniform random variables on $(0,1)$, and write $X_i=F^{-1}(U_i)$, where $F^{-1}(u)=\inf \{x:F(x)\ge u\}$. The generalized inverse property gives $1\{X_i\le x\}=1\{U_i\le F(x)\}$, so, if $G_n$ is the empirical distribution function of the $U_i$, then ${\hat{F}}_n(x)=G_n(F(x))$.

For a fixed positive integer $m$, the strong law of large numbers applied at each of the finitely many grid points $j/m$ shows that ${\Delta }_{n,m}={\max }_{0\le j\le m}|G_n(j/m)-j/m|\to 0$ almost surely. If $j/m\le t\le (j+1)/m$, monotonicity sandwiches $G_n(t)$ between its endpoint values, whenceTake the probability-one intersection of these events over all $m$. On that event the limit superior is at most $1/m$ for every $m$, hence is zero. Therefore ${\sup }_x|G_n(F(x))-F(x)|\to 0$. The coupled sequence has the same joint law as any independent sample from $F$, which proves the asserted almost sure conclusion for the original sample.

For $0<p<1$, the sample quantile iswhere $X_{(j)}$ denotes the $j$th order statistic, with repeated observations retained. Suppose the population median $m=F^{-1}(1/2)$ is unique and $F$ is continuously differentiable in a neighbourhood of $m$, with probability density function $f(m)>0$. ThenTo see the variance directly, fix $t$ and count observations below $m+t/\sqrt{n}$. This count has a binomial distribution with success probability $q_n=1/2+f(m)t/\sqrt{n}+o(n^{-1/2})$. The event that the sample median is below this point is that the count is at least $\lceil n/2\rceil$. The central limit theorem for this binomial triangular array gives a limiting probability $\Phi (2f(m)t)$, exactly the displayed normal distribution. Centering is necessary for stating the limiting distribution; subtracting the deterministic population median has no effect on the asymptotic variance requested in the comparisons.

For the standard normal distribution, $m=0$ and $f(0)=1/\sqrt{2\pi }$, so the sample median has asymptotic variance $\pi /2$ under scaling by $\sqrt{n}$. The central limit theorem gives $\sqrt{n}{\bar{X}}_n\stackrel{d}{\to }N(0,1)$ for the sample mean. Thus the sample median has $\pi /2$ times the asymptotic variance of the sample mean, and its asymptotic relative efficiency against the sample mean is

This is a symmetric beta distribution with parameters $(2,2)$, so its mean and median are both $1/2$. At the median its probability density function is $f(1/2)=3/2$, giving asymptotic variance $1/9$ for the sample median. Direct integration givesConsequently $\sqrt{n}({\bar{X}}_n-1/2)\stackrel{d}{\to }N(0,1/20)$. The sample median has $20/9$ times the asymptotic variance of the sample mean, with asymptotic relative efficiency